Renewal processes, interarrival time distributions

Question

I am dealing with renewal processes recently and I have some questions and I hope you can help me :).

Why interarrival time distributions need to be independent in a renewal process?

Answer 1

A renewal process gets its name from the fact that the process itself renews upon each arrival. This means that the random distribution of future arrival times is exactly the same when observing the process at any given arrival epoch. For this property to hold true, the interarrival times must be independent from one another (and also identically distributed).

The third paragraph here goes into a bit more detail: https://ocw.mit.edu/courses/electrical-engineering-and-computer-science/6-262-discrete-stochastic-processes-spring-2011/course-notes/MIT6_262S11_chap04.pdf

Stationarity

Note that the interarrival times need not be memoryless, so a renewal process is not necessarily stationary. The time until the next arrival can change depending on how long it's been since the last arrival.

You can think of it as being stationary across arrival epochs. The state of the process (the distribution of future arrival times) can vary between arrivals, but will always return to the same original state at each arrival.

If the arrival times are memoryless (exponentially distributed) then the increments will be stationary, meaning the distribution of the time to the next arrival will be constant across time. This arrival time distribution defines the standard Poisson process.